Instead of tilling, why don't you give it a good raking first to see if you can remove the small rocks that are on top. All tilling will do is potentially bring up more rock.

If you don't clear the rocks (pebbles???) before sowing, much of the grass with probably still "take", but you'll have a real problem mowing.

You may also want to bring in more topsoil to spread on top of the existing soil. I'm not real sure why the guy who graded recommended keeping the soil level below the pour, other than to try to make sure that runoff didn't go onto the concrete while it was setting up.

Just some thoughts.--Carl
Atlanta, GA

That is weird! The soil should be right at the height of the concrete and crowned/profiled in the middle so that the water drains out of the yard, away from all buildings, instead of collecting inside the concrete bowl.

Rototilling in preparation for grass is always a bad idea. I realize you can find that recommendation in many places but they are wrong. It leads to a bumpy lawn surface after 3 years of settling.

I would bring in 1.5 more inches of soil at least. Get someone with experience in "finish grading." Usually that is a landscaper who has a tractor and box blade. If he only has a bobcat or skid steer, move on to the next landscaper. Those tools are not good for finish grading. If the yard is as small as you say, you can do this yourself. Smoothing out clods is the hardest part. I prefer sand because it has no clods, but you should probably use the same material you already have. Just pulverize the clods. Make sure the water drains away from all buildings. You might need to build a swale (sort of a ditch) to direct the water out to the street.

If you don't do something about the low soil level, you'll have standing water that will ruin your soil and leave you with wild swamp grasses.